FP1 chapter 3 - coordinate systems

(parametric functions, parabolas and the rectangular hyperbola)

"In this chapter you will be introduced to the parabola and its properties. You will also work with another curve called a rectangular hyperbola."

FP1 section 3.1

Pre-requisite knowledge:

• you need to be confident about substituting and evaluating functions
• it is assumed that you are assured and fluent in rearranging simple linear and quadratic functions
• if you can solve simultaneous linear and quadratic functions as in C1 section 3.3 then you'll be fine
• it is preferable that you have met the notation y = f(x) to express a function
• the book assumes that you know the notation

$\; \; \; \; \; \; \; \; \; \; t \in \mathbb{R}$
. . . . . .which means that the variable t is allowed to be any real number.

• there is an assumption that you can plot and sketch a variety of curves - competence in C1 chapter 2 and C1 chapter 4 will be sufficient

Learning outcomes:

• you will be able to substitute values of t into a simple parametric function x = f(t), y = g(t)
• you will construct a table of values by multiple substitutions t = -3, t = -2, ... , t = 3, etc.
• you will plot the resulting Cartesian graph having evaluated multiple pairs of values ( x , y )
• you will be able to convert suitable parametric functions into their Cartesian form
• you will sketch parametric functions by converting them to Cartesian form and sketching the result

Examples:
Given the parametric function
$x = \text{f} (t), \text{ } y = \text{g} (t), \text{ } t \in \mathbb{R} \text{ where } a > 0 \text{ is a constant.}$
... sketch the curve.

Solution:
Start by creating a table of values for t = -3 to t = 3:

. . . . . . . . . . $t$	. . . . . . . . . . $-3$	. . . . . . . . . . $-2$	. . . . . . . . . . $-1$	. . . . . . . . . . $0$	. . . . . . . . . . $1$	. . . . . . . . . . $2$	. . . . . . . . . . $3$
. . . . . . . . . . $x = at^{2}$	. . . . . . . . . . $9a$	. . . . . . . . . . $4a$	. . . . . . . . . . $a$	. . . . . . . . . . $0$	. . . . . . . . . . $a$	. . . . . . . . . . $4a$	. . . . . . . . . . $9a$
. . . . . . . . . . $y = 2at$	. . . . . . . . . . $-6a$	. . . . . . . . . . $-4a$	. . . . . . . . . . $-2a$	. . . . . . . . . . $0$	. . . . . . . . . . $2a$	. . . . . . . . . . $4a$	. . . . . . . . . . $6a$

then plot the coordinates (x,y) as usual to create your graph.
Here's an animated graph of this function in Desmos. The value of a changes smoothly. You can stop it if you like and explore what happens as you change 'a'.

We don't get this complicated in FP1, but you might enjoy the snowflake emulator I made in Desmos using parametric functions. It's designed to look like the snowflakes that fall in the foreground of the iPad game 'Clash of Clans' at around Christmas.

Why not have a look at the motion of objects in other games and see if you can model their movement with Desmos?

FP1 Exercise 3A

• FP1 Exercise 3A worked solutions:
• 
  
• Questions similar to this exercise:

FP1 section 3.2

Pre-requisite knowledge:

Learning outcomes:

FP1 Exercise 3B

• FP1 Exercise 3B worked solutions:
• 

  FP1 Ex 3B (the parabola).jnt
  

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  • 326 KB
• if you want the pdf, please request it in the comment section below
• Questions similar to this exercise:

FP1 Exercise 3C

• FP1 Exercise 3C worked solutions:
• 

  FP1 Ex 3C (continued) (parabolas and lines).jnt
  

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  • 681 KB
• again the pdf is available on request
• Questions similar to this exercise

FP1 section 3.3